Show line through point that is tangent to f(x)does not exist

[Painguy]

1. show that there is no line through the point (2,7) that is tangent to the parabola y =x^2 +x



2. y-y1=m(x-x1)



3.
$y'=f'(x)=2x+1$
$m1=2x +1$
$m1=2(2) +1 =5$

$m2=((x^2 +x)-7)/(x-2)$

$m1=m2?$
$((x^2 +x)-7)/(x-2)=5$
$x^2-4x+3=0$

$2^2-4(2)+3=/=0$
$-1=0$


I'm thinking that i would compare the slope of the line passing through (2,7) to the general slope of the parabola. The other thing i believe i could do is just plug 2 into y=x^2 +x. that would give me 6 which does not equal 7, but that does not involve using any of the material we're covering in class so that is out of the question.

 

[zooxanthellae]

One way to do it is to pick a random value for our hypothetical line to be tangent to y at: let's call it x1. (If that wasn't clear, we're assuming that we have a line that is tangent to the parabola at x-coordinate x1 and that also contains (2,7). We'll try to derive a contradiction to show that this is impossible). This is basically your idea to compare the slope of a random line through (2,7) to the general parabola slope.

Since our line is tangent to the graph of y at x1, then it has slope 2x1 + 1.
Since our line is tangent to the graph of y at x1, then it contains the point (x1, x1^2 + x1).
So we have a line that contains (x1,x1^2 + x1) and (2,7).
Calculating the slope of such a line using rise/over run, we get a new slope.
By assumption this new slope must equal the old slope, 2x1 + 1.
From there it is easy to set up a quadratic equation and show that it has no real solutions.
This shows that such a line cannot exist!

Your original idea to plug 2 into y does not work because we are only looking for a line that is tangent to y and contains (2,7). The only way your idea would work is if they were asking you to show that (2,7) is not a point on the parabola y.

I hope that is helpful!

 

[Dick]

You are almost there. But the m1=m2 equation you want to try to solve is $((x^2 +x)-7)/(x-2)=2x+1$. Don't substitute x=2 into m1. You don't know the tangent line hits the parabola at x=2. Draw a picture.